In this short note, we generalized the Tangential Cover used in Digital Geometry in order to use very general geometric predicates. We present the required notions of saturated α-paths of a digital curve as well as conservative predicates which indeed cover nearly all geometric digital primitives published so far. The goal of this note is to prove that under a very general situation, the size of the Tangential Cover is linear with the number of points of the input curve. The computation complexity of the Tangential Cover depends on the complexity of incremental recognition of geometric predicates. Moreover, in the discussion, we show that our approach does not rely on connectivity of points as it might be though first.